Glossary

When a sample-rate change makes two different frequencies produce the exact same sequence of numbers, so one is mistaken for (impersonates) the other.

A system whose poles and zeros are conjugate-reciprocal pairs, giving a magnitude response that is flat at every frequency.

How large a signal's value swings -- the A in A\cos(\omega n+\phi), independent of frequency or phase.

The hardware that turns a continuous-time voltage into a sequence of numbers (ADC), or a sequence of numbers back into a voltage (DAC).

How correlated a random signal is with a time-shifted copy of itself, as a function of the shift.

A system is stable when every bounded input is guaranteed to produce a bounded output -- it never blows up on its own.

The s = (2/T)(1-z^-1)/(1+z^-1) substitution mapping an analog prototype to a digital filter with no aliasing, at the cost of nonlinear frequency warping.

The small two-input, two-output computation (one add, one subtract, one multiply) repeated at every stage of the fast Fourier transform.

An analog filter design that is maximally flat in its passband -- the usual starting prototype that chapter 11's bilinear transformation converts into a digital IIR filter.

A specific family of polynomials, T_k(\cos\theta)=\cos(k\theta), whose equal-ripple behavior makes them the basis of two later filter designs.

Flipping the sign of a complex number's imaginary part -- x^* = a - jb for x = a+jb.

A signal defined at every real value of its argument, written x(t).

The formula y[n] = sum_k x[k]h[n-k], computing an LTI system's output from its input and impulse response.

A number between -1 and 1 measuring how much two random quantities tend to move together.

Keeping only every Mth sample of a sequence and throwing the rest away, lowering the effective sample rate by a factor of M.

A signal whose values are given exactly by a formula or rule, with no randomness -- the opposite of a random signal.

An equation defining each output sample from past inputs and (if recursive) past outputs -- the discrete-time equivalent of a differential equation.

An LTI system deliberately designed to reshape a signal's frequency content -- passing some frequencies through and attenuating others.

A function f(x,y) mapping a row and column index to an intensity value -- x[n] with a second index, stored as a 2D array of numbers.

The transform X(e^{jw}) = sum_n x[n]e^{-jwn}, turning a discrete-time sequence into a continuous, 2*pi-periodic frequency-domain function.

A signal defined only at integer-indexed instants, written x[n].

Any change a system makes to a signal's magnitude or phase across frequency, beyond simply scaling it -- the thing a well-designed filter tries to avoid outside its passband.

A signal whose total energy, summed over all time, is finite and nonzero.

The FIR design method (Parks-McClellan) that searches directly for the alternation theorem's own equal-ripple error pattern -- provably minimax-optimal.

A signal is even (symmetric) when x[-n]=x[n], and odd (antisymmetric) when x[-n]=-x[n].

A sequence of the form x[n] = A\alpha^n -- growing, decaying, or (when \alpha is complex) oscillating.

A non-recursive system whose impulse response has finitely many nonzero samples.

An LTI system's DTFT of its own impulse response, H(e^{jw}) -- the factor it multiplies an input's spectrum by.

A recursive system whose impulse response, in principle, never becomes exactly zero.

An LTI system's output when its input is a single unit impulse, written h[n].

Inserting new samples between existing ones (usually zeros, then filtered smooth) to raise the effective sample rate.

A fixed-point recursion getting trapped forever at a small nonzero constant, purely from rounding, instead of decaying to zero the way exact arithmetic guarantees.

A rule is linear when scaling or adding its inputs scales or adds its outputs the same way -- no exponents, products of inputs, or added constants.

A system that is both linear and time-invariant, fully characterized by its impulse response.

How big a number is, ignoring its sign or direction -- |x| for a real number, or the length of the arrow for a complex one.

A stable system whose zeros, not just its poles, all lie strictly inside the unit circle.

Digital frequency measured in radians per sample (\omega) instead of cycles per second -- it stays the same number no matter what the real-world sampling rate was.

Repeated back-and-forth motion around a center value, the way a sinusoid or a resonant filter's output does.

A signal that repeats exactly every N samples, for some smallest positive integer N.

The most direct (and, on its own, unreliable) estimate of a random signal's power spectral density, computed straight from one signal's own DFT magnitude.

Splitting a filter's coefficients by tap index mod M into M shorter sub-filters, letting the downsampler move ahead of the filtering for the identical output at a fraction of the cost.

A signal whose average power, not its total energy, is the finite, meaningful quantity.

The DTFT of a WSS process's own autocorrelation sequence -- a well-defined frequency-domain description no ordinary DTFT of one realization can give.

A two-channel filter-bank design where the highpass branch's impulse response is the lowpass branch's own sign-alternating mirror, making aliasing cancellation an algebraic certainty.

Rounding each sample's continuous amplitude to the nearest of a fixed set of allowed levels, the step every ADC performs after sampling.

Rebuilding a continuous-time signal from its samples, ideally with an ideal lowpass filter -- what a DAC approximates in real hardware.

A lattice filter's own parameter, one per stage, computed from the direct-form coefficients by the step-down recursion.

An ordered list of numbers indexed by an integer, the mathematical object a discrete-time signal is.

An error-feedback converter that pushes quantization noise toward high frequency instead of merely shrinking it, letting a following filter discard most of it.

A function that carries information by varying over time (or another independent variable).

A pure cosine or sine wave, A\cos(\omega n + \phi) -- the single building block every Fourier method decomposes a signal into.

What happens to a signal's DFT when its frequency doesn't land exactly on a bin -- energy spreads across many neighboring bins instead of one.

Two random quantities are independent when knowing one tells you nothing about the other; a distribution describes how likely each possible value is.

H(z), the z-transform of an LTI system's impulse response -- reduces to the frequency response on the unit circle.

The range of frequencies between a filter's passband and stopband, where its response is neither fully passed nor fully blocked.

Variance measures how spread out a random variable's values are around its mean; covariance measures how two random variables spread out together.

A random process whose mean is constant and whose autocorrelation depends only on lag -- the weaker, practically-used version of full stationarity.

The linear filter that provably minimizes mean-square estimation error, characterized by the orthogonality principle.

A tapering function multiplied onto a finite chunk of signal before analysis or FIR design, to control the trade-off between how sharp and how leaky the result is.

A real DAC's cheap reconstruction method -- holding each sample's value constant until the next sample arrives, producing a staircase.